Will Factoring solve the problem?

Algebra Level 3

\[\frac{{\left(x+\dfrac{1}{x}\right)}^6-\left(x^6+\dfrac{1}{x^6}\right)-2}{{\left(x+\dfrac{1}{x}\right)}^3+\left(x^3+\dfrac{1}{x^3}\right)}\]

If \(x\) is a positive real number, then find the minimum value of the expression above.